How to train a neural network for a round based board game?
Great question!  NN is very promising for this type of problem: Giraffe Chess.  Lai's accomplishment was considered to be a pretty big deal, but unfortunately came just a few months before AlphaGo took the spotlight. (It all turned out well, in that Lai was subsequently hired by DeepMind, although not so well for the Giraffe engine;)

I've found Lai's approach to be quite helpful, and it is backed by solid results.



You may want to use "sequential" as opposed to "round based" since sequential is the preferred term in Game Theory and Combinatorial Game Theory, and these are the fields that apply mathematical analysis to games. 

The games you list are further termed "abstract" to distinguish them from modern strategy boardgames, or games in general, which utilize a strong theme and are generally less compact than abstract games in terms of mechanics and elements. This carries the caveat that abstract games are not restricted to sequential games or boardgames, or even games specifically, as in the case of puzzles like Sudoku.

The formal name for this group of games is generally "partisan, sequential, deterministic, perfect information" with the further categorization of Tic-Tac-Toe as "trivial" (solved and easily solvable) and non-trivial (intractable and unsolved) for games like Chess and Go.